This video is especially useful in understanding the conservation of angular momentum and rotational inertia. Rotational inertia is the property of an object to resist changes in motion.
One thing that is really essential to understanding this concept is mass distribution. This depends on how far or close the mass is in consideration to the axis of rotation.
As you can see in the video, the figure skater's speed increases when her arms and legs pull closer into her body and thus closer to her axis of rotation. This gives her a low rotational inertia and a high rotational velocity.  On the other hand, when she spreads her body out, it slows her down and therefore increasing her rotational inertia.
This video does a great job of explaining the changes in rotational inertia and rotational velocity.
 
In the picture above you can see her attempt to pull her mass toward her axis of rotation in order to attain a faster rotational velocity.